HOMOTOPY PERTURBATION TRANSFORM METHOD FOR SOLVING THE PARTIAL AND THE TIME-FRACTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations.


INTRODUCTION
The Parabolic-like and Hyperbolic-like equations can be used to describe wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of Parabolic-like equations and Hyperbolic-like equations. Several authors have solved these linear and nonlinear equations using several methods, for example (ADM), (HPM), (VIM), (2) tt xx yy zz t u g x y z u g x y z u g x y z u t u x y z g x y z u x y z g x y z t For solving these equations, we used the homotopy perturbation transform method (HPTM). These problems have been studied by some researchers by using (ADM) and (HPM) see for example [12]and [18]. We extend our study to the fractional order for solving the problems:

Analysis of Homotopy perturbation transform method (HPTM):
To illustrate the basic idea of (HPTM) [13], we consider a general nonlinear partial differential equation with the initial conditions of the form where D in the second order linear differential operator 2 2 , D t = ∂ ∂ R is the linear differential operator of less order than , D N represents the general nonlinear differential operator and ( ) , g x t is the source term. Taking the Laplace transform on both sides of Eq. (12): Operating with the Laplace inverse on both sides of Eq. (14) gives , G x t represents the term arising from the source term and the prescribed initial conditions. Now, we apply the homotopy perturbation method   (16) and (17) in Eq. (15) we get n n n n n n n n n which is the coupling of the Laplace transform and the homotopy perturbation method He's polynomials. Comparing the coefficient of like powers of p, the following approximations are obtained 1  2  1  1  2   3  1  3  2  2  2   : , , , , , ,

Parabolic-like equation:
Consider the parabolic-like equation in three dimensions of the form: (21) Taking the Laplace transform on both sides of Eq. (21), we get:   Applying the classical perturbation technique, we can assume that the solution can be expressed as a power series in p, as given below: this is a coupling of the Laplace transform and homotopy perturbation methods using He's polynomials. Now, equating the coefficient of corresponding power of p on both sides, the following approximations are obtained as:

Hyperbolic-like equation:
Consider the three dimensional hyperbolic-like equation of the form: (28) tt x x yy zz t u g x y z u g x y z u g x y z u t u x y z g x y z u x y z g x y z t Taking the Laplace transform on both sides of Eq. (28), we get: tt xx yy zz L u L g x y z u g x y z u g x y z u = − + + An application of Eq. (10), yields: xx yy zz u x y z t g t g L s L g u g u g u − − = + − + + Now applying the classical perturbation technique, we can assume that the solution can be expressedas a power series in p , as given below: where the homotopy parameter p , is considered as a small parameter Substituting Eq. (32) in Eq. (31), we get: n n n n n n xx n yy n zz n n n n p u g tg pL s L g p u g p u g p u this is a coupling of the Laplace transform and homotopy perturbation methods using He's polynomials. Now, equating the coefficient of corresponding power of p on both sides, the following approximations are obtained as: Taking the Laplace transform on both sides of Eq. (35), we get: An application of Eq. (10), yields: Applying the inverse Laplace transform on both sides in Eq. (37), we get: By applying the aforesaid homotopy perturbation method, we have: Equating the coefficient of the like power of p on both sides in Eq. (39), we get: , , Finally, we approximate the analytical solution ( ) , u x t , by truncated series: Taking the Laplace transform on both sides of Eq. (42), we get: An application of Eq. (10), yields: Applying the inverse Laplace transform on both sides in Eq. (44), we get: By applying the aforesaid homotopy perturbation method, we have: Equating the coefficient of the like power of p on both sides in Eq. (46), we get , , , Using the iteration formula (47), we obtain Applying the inverse Laplace transform on both sides in Eq. (51), we get: By applying the aforesaid homotopy perturbation method, we have: 24 24 n n n n n n n n xx yy zz n n n n Equating the coefficient of the like power of p on both sides in Eq. (53), weget : Using the iteration formula (54), we obtain

HOMOTOPY PERTURBATION TRANSFORM METHOD FOR SOLVINGTHE PARTIAL 41
Taking the Laplace transform on both sides of Eq. (56), we get: An application of (10) yields: Applying the inverse Laplace transform on both sides of Eq. (58), we get: By applying the aforesaid homotopy perturbation method, we have: Equating the coefficient of the like power of p on both sides in Eq. (60), we get: , Using the iteration formula (61) , , , 1 1 : , , , Using the iteration formula (75), we obtain In order to elucidate the solution procedure of the fractional Laplace homotopy perturbation method [14], we consider the following nonlinear fractional differential equation: x N x is the general nonlinear operator in x , and ( ) , q x t are continuous functions. Now, the methodology consists of applying the Laplace transform first on both sides of (77). Thus, we get: Now, using the differentiation property of the Laplace transform, we have: Operating the inverse Laplace transform on both sides in (79), we get: , G x t , represents the term arising from the source term and the prescribed initial conditions. Now, applying the classical perturbation technique, we can assume that the solution can be expressed as a power series in p , as given below: This is a coupling of the Laplace transform and homotopy perturbation methods using He's polynomials. Now, equating the coefficient of corresponding power of p on both sides, the following approximations are obtained as:

The time-fractional equations of the form (3):
Consider the following time-fractional equation with variable coefficients: This is a coupling of the Laplace transform and homotopy perturbation methods using He's polynomials. Now, equating the coefficient of corresponding power of p on both sides, the following approximations are obtained as: t xx yy zz L D u x y z t L g x y z u g x y z u g x y z u α *

= − + +
An application of Eq. (11), yields: xx yy zz L u x y z t s g x y z s g x y z s L g x y z u g x y z u g x y z u Applying the inverse Laplace transform on both sides in Eq. (95), we get: xx yy zz u x y z g x y z tg x y z L s L g x y z u g x y z u g x y z u Applying the classical perturbation technique, we can assume that the solution can be expressed as a power series in p, as given below: This is a coupling of the Laplace transform and homotopy perturbation methods using He's polynomials. Now, equating the coefficient of corresponding power of p on both sides, the following approximations are obtained as: (100) Taking the Laplace transform on both sides of Eq. (100), we get: An application of Eq. (11), yields: Applying the inverse Laplace transform of both sides in Eq. (102), we get: By applying the aforesaid homotopy perturbation method, we have: Equating the coefficient of the like power of p on both sides in Eq. (104), we get: Using the iteration formula (105)  , , 1 .

ABDELKADER KEHAILI, ALI HAKEM AND ABDELKADER BENALI
By applying the aforesaid homotopy perturbation method, we have: ( ) ( ) p u x y t y Using the iteration formula (113), we obtain ( ) Using the iteration formula (121)